Revealing Optimal Thresholds for Generalized Secretary Problem via Continuous LP: Impacts on Online K-Item Auction and Bipartite K-Matching with Random Arrival Order

Abstract

We consider the general (J, K)-secretary problem, where n totally ordered items arrive in a random order. An algorithm observes the relative merits of arriving items and is allowed to make J selections. The objective is to maximize the expected number of items selected among the K best items.Buchbinder, Jain and Singh proposed a finite linear program (LP) that completely characterizes the problem, but it is difficult to analyze the asymptotic behavior of its optimal solution as n tends to infinity. Instead, we prove a formal connection between the finite model and an infinite model, where there are a countably infinite number of items, each of which has arrival time drawn independently and uniformly from [0, 1].The finite LP extends to a continuous LP, whose complementary slackness conditions reveal an optimal algorithm which involves JK thresholds that play a similar role as the 1/e-threshold in the optimal classical secretary algorithm. In particular, for the case K = 1, the J optimal thresholds have a nice "rational description". Our continuous LP analysis gives a very clear perspective on the problem, and the new insights inspire us to solve two related problems.1. We settle the open problem whether algorithms based only on relative merits can achieve optimal ratio for matroid secretary problems. We show that, for online 2-item auction with random arriving bids (the K-uniform matroid problem with K = 2), an algorithm making decisions based only on relative merits cannot achieve the optimal ratio. This is in contrast with the folklore that, for online 1-item auction, no algorithm can have performance ratio strictly larger than 1/e, which is achievable by an algorithm that considers only relative merits.2. We give a general transformation technique that takes any monotone algorithm (such as threshold algorithms) for the (K, K)-secretary problem, and constructs an algorithm for online bipartite K-matching with random arrival order that has at least the same performance guarantee.